# Full text of "Treatise On Analysis Vol-Ii"

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```5    PRODUCTS OF UNIFORMIZABLE SPACES       15

where At- and Et are open sets in Et (/ = 1, 2). This topology £) is called the
product of the topologies on EA and E2 , and the set E endowed with this
topology is called the product of the topological spaces El9 E2 . If x = (xl9 x2)
is any point of E, the sets Vx x V2 (where V£ runs through a fundamental
system of neighborhoods of xi9 for/ = 1, 2) form a fundamental system of
neighborhoods of x in E (3.9.3). From this it follows that the continuity
criterion (3.20.4) is valid for the product of two arbitrary topological spaces.

The relation AJ x A2 = Ax x A2 (where AA c Ex and A2 c E2) also
remains true; for if (al9 #2) e E and if V£ is a neighborhood of #,- in E,-
(/ = 1, 2), the set (Vj x V2) n (A1 x A2) = (Vj n AJ x (V2 n A2) is empty
if and only if one of the sets Vf n A,- is empty.

For each a1eE1, the set ({^ } x E2) n (Aj x A2) is either empty or equal
to {a^} x A2 , and therefore the mapping JC2i— K#i» ^2) *s a homeomorphism
of E2 onto the subspace {al } x E2 of E. Likewise, if a2 e E2 , the mapping
X1\-^(xl9 a2) is a homeomorphism of EJ onto the subspace E1 x {a2} of E.
From this it follows immediately that the propositions (3.20.12) and (3.20.1 3),
and the continuity criteria (3.20.14) and (3.20.15), are valid in general.

If Ex and E2 are Hausdorffthtn so is E1 x E2 . For if x = (xl9 x2) and
y = (yt, y2) are distinct, then either jq ^ y1 or x2 ^ y2 . If the latter, there is
a neighborhood U of x2 and a neighborhood V of y2 in E2 which do not
intersect, and then EJ x U and Et x V are disjoint neighborhoods of x and y,
respectively, in E.

If E2 = El9 the canonical symmetry (xt, x2)t-*(x2 , x^) is a homeomorphism
of E! x E! onto itself, and is equal to its inverse.

The product of any finite number of topological spaces is defined in the
same way. It follows immediately from the definition that the canonical
"associativity" mappings such as (E1 x E2) x E3 -^El x (E2 x E3) are
homeomorphisms.

We shall be concerned especially with the case in which Ej and E2 are
uniformizable. In this case the product space E is also uniformizable. To be
precise, let (d^)^GL9 (fl^^eM be families of pseudo-distances which define
the topologies of Ei9 E2 respectively, and let

, J') = d^pr.x, pr^),       ef\x, y) = d?\pr2 x, pr2 y).

Then the e^ and ^2) are pseudo-distances on E, and the definition of neigh-
borhoods given above shows that the family of pseudo-distances e^ and
e(*\ where X e L and \JL € M, defines the product topology on E.

This leads to a generalization of the notion of a product to arbitrary
(not necessarily finite) families of topological spaces. We shall restrict our-
selves to uniformizable spaces.merable dense subset. But there exists no denumerable basis of
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